Measurements are the interface between the intricate quantum world and the perceivable macroscopic classical world. A measurement associates to a quantum state a classical attribute. However, quantum phenomena, such as superposition, entanglement and non-commutativity contribute to uncertainty in the measurement outcomes. A key concern, from an information-theoretic standpoint, is to quantify the amount of “relevant information” conveyed by a measurement about a quantum state.
Winter’s measurement compression theorem (as elaborated in ) quantifies the “relevant information” as the amount of resources needed to simulate the output of a quantum measurement applied to a given state. Imagine that an agent (Alice) performs a measurement on a quantum state and sends a set of classical bits to a receiver (Bob). Bob intends to faithfully recover the outcomes of Alice’s measurements without having access to . The measurement compression theorem states that at least quantum mutual information () amount of classical information and conditional entropy () amount of common shared randomness are needed to obtain a faithful simulation.
The measurement compression theorem finds its applications in several paradigms including local purity distillation  and private classical communication over quantum channels . This theorem was later used by Datta, et al.  to develop a quantum-to-classical rate-distortion theory. The problem involved lossy compression of a quantum information source into classical bits, with the task of compression performed by applying a measurement on the source. In essence, the objective of the problem was to minimize the storage of the classical outputs resulting from the measurement while ensuring sufficient reliability so as to be able to recover the quantum state (from classical bits) within a fixed level of distortion from the original quantum source. To achieve this, the authors in  advocated the use of measurement compression protocol and subsequently characterized the so called rate-distortion function in terms of single-letter quantum mutual information quantities. The authors further established that by employing a naive approach of measuring individual output of the quantum source, and then applying Shannon’s rate-distortion theory to compress the classical data obtained is insufficient to achieve optimal rates.
In this work, we seek to quantify “relevant information” for quantum measurements performed in a distributed fashion. In this setting, as shown in Fig. 1, a composite bipartite quantum system is made available at two separate agents, named Alice and Bob. Alice and Bob have access only to sub-system and , respectively. Two separate measurements, one for each sub-system, are performed in a distributed fashion with no communication taking place between Alice and Bob. Imagine that there is a third party (named Eve) who tries to simulate the action of the measurements without any access to the quantum systems. To achieve this objective, Alice and Bob send classical bits to Eve at rate and , respectively. Eve on receiving these pairs of classical bits from Alice and Bob wishes to reconstruct the joint quantum state using a classical-to-quantum channel. The reconstruction has to satisfy a fidelity constraint characterized using a distortion observable or a norm-distance.
A naive strategy is to apply Winter’s measurement theorem  to compress each individual measurement and separately into and . As a result, faithful simulation of by is possible when at least classical bits of communication and bits of common randomness are available between Alice and Eve. Similarly, a faithful simulation of by is possible with classical bits of communication and bits of common randomness between Eve and Bob. However, we can further reduce the amount of classical communication by exploiting the statistical correlations between Alice’s and Bob’s measurement outcomes.
As one of the major contributions of this work, we develop a method to simulate Alice’s and Bob’s measurements with lower number of communication bits. In addition, we characterize a set of sufficient communication and common randomness rates in terms of single-letter quantum information quantities (Theorem 2). To prove this theorem, we develop two techniques: 1) binning of quantum measurements, and 2) mutual packing lemma for distributed quantum measurements. These techniques can be viewed as the quantum counterpart of their classical analogues . The idea of binning in quantum setting has been used in  and  for quantum data compression involving side information (similar to Slepian-Wolf problem). However, in this paper we introduce a novel binning technique for measurements which is different from these works. The binning in this work is used to construct measurements for Alice and Bob with fewer outcomes compared to the above individual measurements, i.e., and . The mutual packing lemma is used to ensure that the binned measurements performs as good as these individual measurements.
Secondly, we use our results on the simulation of distributed measurements to develop a distributed quantum-to-classical rate distortion theory (Theorem 3). For the achievability part, we characterize a rate region analogous to Berger-Tung’s  in terms of single-letter quantum mutual information quantities. Further, we can show that, as in the classical setting, the n-letter regularization of this rate region is optimal.
The paper is organized as follows: In Section II, basic definitions and formulations are provided. Section III contains our results on distributed simulation of quantum measurements. In Section IV, we derive the quantum counterpart of Berger-Tung rate region for quantum-classical distributed source coding. Finally, Section V concludes the paper.
We here establish all our notations, briefly state few necessary definitions, and also provide Winter’s theorem on measurement compression. Let denote the algebra of all linear operators acting on a finite dimensional Hilbert space . Further, let denote the set of positive operators of unit trace acting on . By denote the identity operator. The trace distance between two operator and is defined as , where for any operator we define . The Von Neumann entropy of a density operator is denoted by . The quantum mutual information and conditional entropy for a bipartite density operator are defined, respectively, as
A positive-operator valued measure (POVM) is a collection of of positive operators that form a resolution of the identity:
Let denote a purification of a density operator . Given a POVM acting on define
Consider two POVMs and acting on and , respectively. Define as a the collection of all observables of the form . With this definition, is a POVM acting on . By denote the -fold tensor product of the POVM with itself.
Consider a POVM with classical outputs . Given a mapping , define as a new POVM with observables but with classical outputs where . For this POVM equation (1) can be written as
Ii-a Quantum Information Source
Consider a family of quantum states acting on a Hilbert space
. For each state assign a priori probability. We denote such a setup by the ensemble . For such an ensemble, a quantum source is a sequence of states each equal to with probability . Each realization of the source, after generations of states, is represented by , where
is a vector with elements in. Let , then the average density operator of the source after generations is .
Ii-B Measurement Compression Theorem
Here, we provide a brief overview of the measurement compression theorem .
Definition 1 (Faithful simulation).
Given a POVM and a density operator acting on a Hilbert space , a POVM acting on is -faithful if for the following holds:
where are the observables of .
 For , a density operator and a POVM , there exist a collection of POVMs for , each acting on , and having at most outcomes where
such that is -faithful.
Iii Approximation of Distributed POVMs
We provide our extension to the Winter’s measurement compression protocol for a distributed setting. Consider a bipartite composite quantum system represented by Hilbert Space . Let be a quantum information source on . Imagine that three parties, traditionally named Alice, Bob and Eve, are trying to collectively implement two measurements, one applied to each sub-system. Eve has no access to the quantum system; while Alice and Bob have access to sub-system and , respectively. Alice and Bob perform a measurement and on sub-systems and , respectively. The measurements are performed in a distributed fashion with no communication taking place between Alice and Bob. In this context, the overall measurement is characterized by the tensor product measurement . The objective of Eve is to reconstruct an asymptotically faithful simulation of when acted on . For that, Alice and Bob send a number of classical bits to Eve. Then, Eve applies a decoding map to the received bits and reconstructs the original measurement outcomes. The design objective is to minimize the amount of the classical bits that Eve needs to simulate the measurements.
We consider a special class of quantum information sources. In particular, the focus of this section is on non-entangled quantum information sources defined as in the following.
A non-entangled quantum state is characterized by an ensemble where
In what follows, we formulate the problem.
For a given Hilbert space , a distributed protocol with classical communication rates and common randomness rate is characterized by a collection of POVM-pairs , acting on and with at most and outcomes.
In the above definition, determines the amount of classical bits communicated from Alice and Bob to Eve. The amount of common randomness is bits and can be viewed as the common randomness bits distributed among the parties. In the following, we define a measure for faithful simulation.
A distributed protocol with POVM-pairs is -faithful for simulation of the POVM on the source , if there exist a collection of mappings such that the average POVM is -faithful according to Definition 1.
In the above definition, the mappings represent the action of Eve on the received classical bits.
Let denote a non-entangled quantum information source measured by a POVM of the form . For any and sufficiently large , there exists an -faithful distributed protocol with classical communication rates and common randomness rate provided that
where the mutual information and conditional entropy terms are calculated for the state
where is a purification of and denote the observables of .
The proof is given in Appendix. ∎
Fig. 2 demonstrates the region in Theorem 2 in terms of the quantum information quantities. It also shows the gains achieved by employing such an approach as opposed to independently compressing the two sources and .
As for the converse, one can extend the single letter characterization to an n-letter regularized formulation and provide an optimal rate region with a converse to its achievability. Define as the set of all rates for which there exist such that
We can show that is the optimal rate-region for this problem. Note that characterizing the optimal rate region in terms of single letter information quantities is still an open problem even in classical setting.
Iii-a Proof Techniques
Binning for POVMs
We introduce a quantum-counterpart of the classical binning technique used to prove Theorem 2. Here, we describe this technique.
Consider a POVM with observables . Given for which is divisible, partition into equal bins and for each , let denote the bin. The binned POVM is given by the collection of operators where is defined as
Using the fact that are self-adjoint and positive and , (which is because is a POVM); it follows that is a valid POVM.
Mutual Packing Lemma for POVMs
Another technique used to prove Theorem 2 is a quantum version of mutual packing lemma. In what follows, we describe the mutual packing lemma for quantum measurements. For a Hilbert Space consider a POVM of the form , where are two POVMs each acting on one sub-system. The observables for and are denoted, respectively, by and , where and
are finite sets. Fix a joint-distributionon the set of all outcomes . For each , let be a random sequence generated according to . Similarly, let be a random sequence distributed according to , where . Suppose ’s and ’s are independent. Define the following random observables:
where and .
For any and sufficiently large , with high probability
provided that .
From the triangle-inequality and the definition of and , the norm in the lemma does not exceed the following
where the last inequality holds since . The proof completes from the classical mutual packing lemma. ∎
Iv Q-C Distributed Rate Distortion Theory
As an application to the above theorem on faithful simulation of distributed measurements (Theorem 2), we investigate the distributed extension of quantum-to-classical rate distortion coding . This problem is a quantum counterpart of the classical distributed source coding. In this setting, many copies of a bipartite quantum information source are generated. Alice and Bob have access the partial trace of the copies related to and . They perform measurements on their marginalized source. The goal is to generate a classical description of the source so that a third party, namely Eve, would be able to reconstruct the source within a distortion threshold. First, we formulate the problem in the following.
A q-c source coding setup is characterized by an input Hilbert space , an information source acting on , a reconstruction Hilbert space , and a distortion observable acting on such that it is non-negative and bounded, i.e., for some .
The distortion defined for the action of a POVM on a state is measured by
For n-letter distortion, we use average distortion observable defined as
This gives the n-letter average distortion as
where is the measurement map acting on the source state and is the purification of the state .
The authors in  studied the point-to-point version of the above formulation. They considered a special distortion observable of the form
where acts on the reference Hilbert space and is the reconstruction alphabet. In this paper, we allow to be any non-negative and bounded operator acting on the appropriate Hilbert spaces. Moreover, we allow for the use of any c-q mapping as a decoder.
In the distributed setting, we define the distortion observable as any operator acting on the Hilbert space . In what follows we provide a definition for a distributed quantum-classical compression code.
An Quantum-to-Classical (q-c) code is defined by POVMs and with and outcomes, respectively, and a mapping
Given a source , a pair of rates () and a distortion are said to be achievable if for all and sufficiently large n, there exists an q-c code such that
For a given distortion threshold , the set of all rate pairs such that is achievable is called as the rate distortion region.
For a bipartite quantum source , a distortion observable and any given distortion , the q-c rate distortion region is the union of all rate-pairs that satisfy
where the union is taken over all POVMS with .
The proof follows from Theorem 2. Fix POVMs as in the statement of the theorem. Hence,
According to Theorem 2, there exists a set of POVMs each pair having at most outcomes, where
and a mapping such that is a faithful simulation of , where
Therefore, from Definition 1, and large enough , the following condition is satisfied:
Suppose represents the outcomes of the POVMs . Then the mapping can viewed as the classical to quantum decoder for such measurements. In particular the decoder is .
With this notation, it suffixes to show the following bounds
One can observe that the rate-region in Theorem 3 matches exactly with the classical Berger-Tung region when is a mixed state of a collection of orthogonal pure states. Note that the rate-region is an inner bound for the set of all achievable rates. The single-letter characterization of the set of achievable rates is still an open problem even in the classical setting.
We established a distributed extension of Winter’s measurement compression theory. A set of communication rate-pairs and common randomness rate is characterized for faithful simulation of distributed measurements. We further investigated distributed quantum-to-classical rate-distortion theory and derived a quantum counterpart of Berger-Tung rate-region.
-a Proof of the Theorem
Suppose the observables of and are denoted by and , respectively, where are two finite subsets. The canonical ensembles corresponding to and are defined as
where and . Note that
is a joint probability distribution onwith and as the marginals. With this notation, corresponding to each of the probability distributions, we can associate a -typical set. Let us denote , and as the -typical sets defined for , and , respectively.
Let and denote the -typical projectors (as in ) for marginal density operators and , respectively. Also, for any and , let and denote the conditional typical projectors (as in ) for the canonical ensembles and , respectively. For each and define
where and .
Let and be random sequences generated independently and according to
and for any and . Here and are chosen such that . With the notation above, define and , where the expectation is taken with respect to and , respectively. Let and be the projector onto the subspace spanned by the eigen-states of and
corresponding to eigenvalues that are larger thanand , respectively. Lastly, define
-B Construction of Random POVMs
In what follows, we construct two random POVMs one for each encoder. Fix a positive integer . For each , randomly and independently select and sequences according to and (as in (7) and (8)), respectively. Let represent the randomly selected sequences for each , where and . Construct observables
where is a parameter to be determined. Then, for each construct and as in the following
As a first step, one can show that with probability sufficiently close to one, and form sub-POVMs for all . More precisely the following Lemma holds.
For any , as in (12), and any , there is such that for all the collection of observables and form sub-POVMs for all with probability atleast , provided that
The proof uses a similar argument as in the proof of Theorem 2 in . Hence it is omitted. ∎
-C Binning of POVMs
We introduce the quantum counterpart of the so-called binning technique which has been widely used in the context of classical distributed source coding. Fix binning rates . Partition the sets